Terence Tao

The thing about math is it's not just about taking a technique that is going to work and applying it, but you need to not take the techniques that don't work.

And for the problems that are really hard, often there are dozens of ways that you might think might apply to solve the problem.

But it's only after a lot of experience that you realize there's no way that these methods are going to work.

So having these counterexamples for nearby problems kind of rules out, it saves you a lot of time because you're not wasting energy on things that you now know cannot possibly ever work.

Right.

Yeah.

So the key phenomenon that my technique exploits is what's called supercriticality.

So in partial differential equations, often these equations are like a tug of war between different forces.

So in Navier-Stokes, there's the dissipation force coming from viscosity, and it's very well understood.

It's linear.

It calms things down.

If viscosity was all there was, then nothing bad would ever happen.

But there's also transport, that energy in one location of space can get transported because the fluid is in motion to other locations.

And that's a nonlinear effect, and that causes all the problems.

So there are these two competing terms in the Navier-Stokes equation, the dissipation term and the transport term.

If the dissipation term dominates, if it's large, then basically you get regularity.

And if the transport term dominates, then we don't know what's going on.

It's a very nonlinear situation.

It's unpredictable.

It's turbulent.